Abstract : The notion of symmetry classically deﬁned for hyperbolic systems of conservation laws is extended to the case of evolution equations of conservative form for which the ﬂux function can bean operator. We explain how such a symmetrization can work from a general point of view using an extension of the classical Godunov structure. We then apply it to the Green–Naghdi type equations which are a dispersive extension of the hyperbolic shallow-water equations. In fact, in the case of these equations, the general Godunov structure of the system is obtained from its Hamiltonian structure.